A radio frequency (RF) receiver can downconvert a modulated signal at a carrier frequency to a baseband frequency. During the downconversion of the modulated signal, certain receiver impairments are introduced due to non-ideal behavior caused by, for example, component process-voltage-temperature (PVT) variations. The receiver impairments degrade system performance. Receiver impairments introduced by a homodyne (direct-conversion) receiver include carrier frequency offset (CFO), IQ mismatch, DC offset, phase noise, etc. The receiver impairments can be minimized via signal processing.
FIG. 1 illustrates an ideal downconversion circuit 10, operating based on local oscillator (LO) signals (cosine and sine signals) that are 90° out-of-phase from one another. The downconversion circuit 10 includes a first mixer 12, a second 14, a first low pass filters (LPF) 16 and a second LPF 18. The mixers 12, 14 receive an RF modulated signal xRF(t). The modulated signal xRF(t) includes a baseband signal x(t) that is shifted to a carrier frequency (ωc). The modulated signal xRF(t) may be represented by equation 1, and the baseband signal may be represented by equation 2.xRF(t)=xI(t)cos ωct−xQ(t)sin ωct  (1)x(t)=xI(t)+jxQ(t)  (2)
The mixers 12, 14 receive and multiply the modulated signal xRF(t) by local oscillator signals 2cos ωct and 2sin ωct to generate I and Q baseband signal components xI(t)cos ωct and xQ(t)sin ωct, respectively, and some unwanted high frequency components. This may be referred to as quadrature mixing. xI(t) and xQ(t) include I and Q baseband components, respectively, and ωc is the carrier frequency. The filters 16, 18 filter out the high frequency components from the outputs of the mixers 16, 18.
Deviation from 90° difference in phase between the I and Q baseband signal components and difference in gains between the I and Q baseband signal components results in distortion and degraded quality of the resulting baseband signal x(t). Deviation of the local oscillator (LO) frequency (ωc) from the actual frequency of the carrier signal results in carrier frequency offset (CFO) and also degraded quality of the resulting baseband signal x(t).
FIG. 2 illustrates another downconversion circuit 20 operating based on local oscillator signals that are both gain mismatched and phase mismatched. The gain mismatch is modeled by the variable ε, and the phase mismatch is modeled by the variable θ. The downconversion circuit 20 includes a first mixer 22, a second mixer 24, a first low pass filter 26, and a second low pass filter 28. The mixers 22,24 receive and multiply a modulated signal xRF(t) by local oscillator signals
      2    ⁢          (              1        +                  ɛ          2                    )        ⁢          cos      ⁡              (                                            ω              c                        ⁢            t                    +                      θ            2                          )              ,            -      2        ⁢          (              1        -                  ɛ          2                    )        ⁢          sin      ⁡              (                                            ω              c                        ⁢            t                    -                      θ            2                          )            that are gain mismatched and phase mismatched, resulting in non-ideal downconversion.
The non-ideal downconverson results in mixing of I and Q components of the corresponding baseband signal components to provide received signal components wI(t) and wQ(t) having IQ mismatch. The received signal components with wI(t) and wQ(t) may be represented by equations 3 and 4, where ε and θ are respectively gain and phase mismatch parameters.
                                          w            I                    ⁡                      (            t            )                          =                              (                          1              +                              ɛ                2                                      )                    ⁢                      (                                                                                x                    I                                    ⁡                                      (                    t                    )                                                  ⁢                cos                ⁢                                  θ                  2                                            +                                                                    x                    Q                                    ⁡                                      (                    t                    )                                                  ⁢                sin                ⁢                                  θ                  2                                                      )                                              (        3        )            
                                          w            Q                    ⁡                      (            t            )                          =                              (                          1              -                              ɛ                2                                      )                    ⁢                      (                                                                                x                    I                                    ⁡                                      (                    t                    )                                                  ⁢                sin                ⁢                                  θ                  2                                            +                                                                    x                    Q                                    ⁡                                      (                    t                    )                                                  ⁢                cos                ⁢                                  θ                  2                                                      )                                              (        4        )            In equation 3, wI(t) includes a Q component (xQ(t)). In equation 4, wQ(t) includes an I component (xI(t)). Equations 3 and 4 can be represented in complex form, as shown by equation (5), where equation 6 provides the received signal w(t) with IQ mismatch, equation 7 provides the baseband equivalent received signal prior to IQ mismatch x(t), and equation 8 provides a corresponding complex IQ mismatch parameter a.
                              w          ⁡                      (            t            )                          =                                                            (                                                      cos                    ⁢                                          θ                      2                                                        -                                      j                    ⁢                                          ɛ                      2                                        ⁢                    sin                    ⁢                                          θ                      2                                                                      )                            ⁢                              x                ⁡                                  (                  t                  )                                                      +                                          (                                                                            ɛ                      2                                        ⁢                    cos                    ⁢                                          θ                      2                                                        +                                      j                    ⁢                                                                                  ⁢                    sin                    ⁢                                          θ                      2                                                                      )                            ⁢                                                x                  *                                ⁡                                  (                  t                  )                                                              ≈                                    x              ⁡                              (                t                )                                      +                                          a                ⁡                                  (                  m                  )                                            ·                                                x                  *                                ⁡                                  (                  t                  )                                                                                        (        5        )            w(t)=wI(t)+jwQ(t)  (6)x(t)=xI(t)+jxQ(t)  (7)
                    a        =                              ɛ            2                    +                      j            ⁢                          θ              2                                                          (        8        )            
A final approximation in equation 5 for the received signal w(t) may be obtained by assuming that the gain and phase mismatch parameters ε and θ, respectively, are small values. In the frequency domain, the received signal w(t) may be represented by equation 9, which shows that IQ mismatch introduces an image a·X*(−f) into the corresponding signal spectrum.W(f)=X(f)+a·X*(−f)  (9)The image a·X*(−f) is a frequency byproduct of the actual signal X(f).